Matsubara formalism

The zero-temperature formalism of many-body perturbation theory breaks down for metals (systems with zero energy band-gap) as pointed out by Kohn and Luttinger. This conundrum is lifted by considering diagrammatic perturbation theory at finite temperature $T>0$, which may be understood by an analytical continuation of the real-time $t$ to the imaginary time axis $-i\tau$. Matsubara has shown that this Wick rotation in time $t\to -i\tau$ reveals an intriguing connection to the inverse temperature $\beta =1/T$ of the system. More precisely, Matsubara has shown that all terms in perturbation theory at finite temperature can be expressed as integrals of imaginary time quantities (such as the polarizability $\chi (-i\tau )$) over the fundamental interval $-\beta \leq \tau \leq \beta$.

As a consequence, one decomposes imaginary time quantities into a Fourier series with period $\beta$ that determines the spacing of the Fourier modes. For instance the imaginary polarizability can be written as

$\chi (-i\tau )={\frac {1}{\beta }}\sum _{m=-\infty }^{\infty }{\tilde {\chi }}(i\nu _{m})e^{-i\nu _{m}\tau },\quad \nu _{m}={\frac {2m}{\beta }}\pi$

and the corresponding random-phase approximation of the correlation energy at finite temperature becomes a series over (in this case, bosonic) Matsubara frequencies

$\Omega _{c}^{\rm {RPA}}={\frac {1}{2}}{\frac {1}{\beta }}\sum _{m=-\infty }^{\infty }{\rm {Tr}}\left\lbrace \ln \left[1-{\tilde {\chi }}(i\nu _{m})V\right]-{\tilde {\chi }}(i\nu _{m})V\right\rbrace ,\quad \nu _{m}={\frac {2m}{\beta }}\pi$

The Matsubara formalism has the advantage that all contributions to the Green's function and the polarizability are mathematically well-defined, including contributions from states close to the chemical potential $\epsilon _{n{\bf {k}}}\approx \mu$, such that Matsubara series also converge for metallic systems.

Although formally convenient, the Matsubara series converges poorly with the number of considered terms in practice. VASP, therefore, uses a compressed representation of the Fourier modes by employing the Minimax-Isometry method. This approach converges exponentially with the number of considered frequency points.